Turn off motion blur in cuts
Here is my solution: It requires some manual labor.
Update: Only works for Cycles
I will be saying "Ghost Effect" a lot in this answer. The image is what I mean by that.


First, locate the property titled "Position" in the Motion Blur panel. The position property defines how the Motion Blur is calculated:
Start on Frame: The Motion Blur is calculated through the current frame and the frame right after it.
Center on Frame: Calculated through frame - 1, frame, and frame + 1.
End on Frame: Calculated through frame - 1 and frame.
With these three settings, you can remove the "Ghost Effect" of motion blur with this procedure: Let's say that the cut is happening on frame and frame + 1. If you use the default Center on Frame, both frame and frame + 1 will have the ghost effect, because the Motion Blur calculates with neighboring frames. To fix this, keep it on Center on Frame, go to frame - 1, hover over the property with your mouse, and press 'i' to add a keyframe. Then, go to frame and change it to End on Frame and add a keyframe. Then, go to frame + 1 and add a keyframe on Start on Frame. Last, add a keyframe on Center on Frame for frame + 2. In total, you will need to add 4 keyframes.
Here's an explanation of what this is doing:
On frames that are not in the cut, the Motion Blur computes on Center on Frame. When it reaches the first frame of the cut, it performs on End on Frame. End on Frame computes with frame - 1 and frame, as stated above. Because the cut happens on frame and frame + 1, the Motion Blur does not reach the cut, and will eliminate the "Ghost Effect". This can also be applied to the next frames.